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\title{\vspace{-2cm} \textbf{第十二次课堂作业}}

\author{邵柯欣 \\学号：3200103310 \\课程名称：数据科学的数学基础}

\date{\today}

\begin{document}
\maketitle
\section{7.1}
\subsection{1}
the third right singular value:
\[
\begin{bmatrix}
   -0.1559\\
    0.4814\\
    0.1814\\
   -0.7085\\
   -0.3543\\
   -0.2890\\
\end{bmatrix}
\]
\subsection{2}
the second singular value:
\[
    0.1994
\]
\subsection{3}
the fourth left singular value:
\[
\begin{bmatrix}
    0.1255\\
    0.0156\\
    0.0436\\
    0.2362\\
    0.2106\\
    0.3250\\
   -0.2208\\
   -0.1704\\
   -0.0099\\
    0.1839\\
   -0.3154\\
    0.1080\\
    0.1406\\
    0.1384\\
   -0.1896\\
   -0.0855\\
    0.0532\\
    0.0863\\
   -0.1979\\
    0.0593\\
    0.1471\\
    0.2691\\
    0.1068\\
   -0.1467\\
   -0.0916\\
    0.3265\\
   -0.3989\\
    0.0989\\
   -0.0148\\
   -0.0856\\
\end{bmatrix}
\]
\subsection{4}
the rank of A is 6.
\subsection{5}
\[
\|A - A_K\|^2_F = 0.0016.
\]
\subsection{6}
\[
\|A - A_K\|^2_F = 0.0016.
\]
\subsection{7}
\[
\|A - \pi_B(A)\|^2_F = 1.8367.
\]
\subsection{8}
\[
\|A - \pi_B(A)\|^2_2 = 1.7171.
\]
\section{7.5}
\subsection{1}
Proof:\\
$\because \quad d(1,3) = 3.4, d(3,6) = 3.3, d(1,6) = 7.7. $\\
$\therefore \quad d(1,6) > d(1,3) + d(3,6).$\\
$\therefore \quad$ D is not a metric.
\subsection{2}
Solution:\\
The coordinates is:
\[
\begin{bmatrix}
   -3.5030  &       0\\
   -3.8424  &       0\\
   -0.8664  &       0\\
    1.3332  &       0\\
    2.6909  &       0\\
    4.1877  &       0\\
\end{bmatrix}
\]
\subsection{3}
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.8]{./hw12.png}
\end{figure}
\section{7.10}
\subsection{1}
\[
Q =
\begin{bmatrix}
   -0.0866  &  0.0523        & 0        & 0        & 0        & 0\\
   -0.0647  &  0.0125        & 0        & 0        & 0        & 0\\
    0.3229  &  0.0291        & 0        & 0        & 0        & 0\\
   -0.1343  &  0.0240        & 0        & 0        & 0        & 0\\
    0.0112  & -0.0005        & 0        & 0        & 0        & 0\\
    0.0704  & -0.0557        & 0        & 0        & 0        & 0\\
   -0.0284  & -0.0535        & 0        & 0        & 0        & 0\\
   -0.0615  &  0.0019        & 0        & 0        & 0        & 0\\
   -0.0119  &  0.0218        & 0        & 0        & 0        & 0\\
   -0.1641  &  0.0266        & 0        & 0        & 0        & 0\\
   -0.0601  & -0.0090        & 0        & 0        & 0        & 0\\
   -0.3691  & -0.0330        & 0        & 0        & 0        & 0\\
   -0.1693  & -0.0304        & 0        & 0        & 0        & 0\\
   -0.2329  &  0.0161        & 0        & 0        & 0        & 0\\
    0.1322  &  0.0049        & 0        & 0        & 0        & 0\\
    0.1674  & -0.0150        & 0        & 0        & 0        & 0\\
    0.1716  & -0.0788        & 0        & 0        & 0        & 0\\
    0.1286  & -0.0269        & 0        & 0        & 0        & 0\\
   -0.0499  &  0.0280        & 0        & 0        & 0        & 0\\
   -0.0650  &  0.0500        & 0        & 0        & 0        & 0\\
    0.1009  & -0.0935        & 0        & 0        & 0        & 0\\
   -0.0320  &  0.0654        & 0        & 0        & 0        & 0\\
    0.0532  & -0.0060        & 0        & 0        & 0        & 0\\
    0.0261  &  0.0273        & 0        & 0        & 0        & 0\\
    0.5164  &  0.0316        & 0        & 0        & 0        & 0\\
    0.0926  & -0.0086        & 0        & 0        & 0        & 0\\
   -0.2830  & -0.0166        & 0        & 0        & 0        & 0\\
   -0.1317  & -0.0096        & 0        & 0        & 0        & 0\\
   -0.3182  & -0.0145        & 0        & 0        & 0        & 0\\
   -0.1401  & -0.0052        & 0        & 0        & 0        & 0\\
\end{bmatrix}
\]
\end{document}
